Analytic Number Theory Exponent Database

Bibliography

1

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2

R. J. Backlund. über die Nullstellen der Riemannschen Zetafunktion. Acta Math., 41(1):345–375, 1916.

3

R. C. Baker. The Brun-Titchmarsh theorem. J. Number Theory, 56(2):343–365, 1996.

4

R. C. Baker and G. Harman. The Brun-Titchmarsh theorem on average. In Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), volume 138 of Progr. Math., pages 39–103. Birkhäuser Boston, Boston, MA, 1996.

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R. C. Baker and G. Harman. The difference between consecutive primes. Proc. London Math. Soc. (3), 72(2):261–280, 1996.

6

R. C. Baker, G. Harman, and J. Pintz. The exceptional set for Goldbach’s problem in short intervals. In Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), volume 237 of London Math. Soc. Lecture Note Ser., pages 1–54. Cambridge Univ. Press, Cambridge, 1997.

7

R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83(3):532–562, 2001.

8

S. Baluyot. On the zeros of Riemann’s zeta-function. PhD thesis, University of Rochester, 2017.

9

Chiara Bellotti. Explicit zero density estimate near unity, 2023.

10

Chiara Bellotti. An explicit log-free zero density estimate for the riemann zeta-function, 2024.

11

Chiara Bellotti and Andrew Yang. On the generalised Dirichlet divisor problem. Bulletin of the London Mathematical Society, 56(5):1859–1878, May 2024.

12

M. Bennett. Fractional parts of powers of rational numbers. Mathematical Proceedings of the Cambridge Philosophical Society, 114(2):191–201, 1993.

13

E. Bombieri and H. Iwaniec. On the order of ζ(12+it). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Ser. 4, 13(3):449–472, 1986.

14

Enrico Bombieri. Le grand crible dans la théorie analytique des nombres, volume No. 18 of Astérisque. Société Mathématique de France, Paris, 1974. Avec une sommaire en anglais.

15

J. Bourgain. Remarks on Montgomery’s conjectures on Dirichlet sums. In Geometric aspects of functional analysis (1989–90), volume 1469 of Lecture Notes in Math., pages 153–165. Springer, Berlin, 1991.

16

J. Bourgain. On the distribution of Dirichlet sums. II. In Number theory for the millennium, I (Urbana, IL, 2000), pages 87–109. A K Peters, Natick, MA, 2002.

17

J. Bourgain. Decoupling, exponential sums and the Riemann zeta function. Journal of the American Mathematical Society, 30(1):205–224, January 2017.

18

J. Bourgain and M. Z. Garaev. Kloosterman sums in residue rings. Acta Arith., 164(1):43–64, 2014.

19

Jean Bourgain. Remarks on Halasz-Montgomery Type Inequalities. In J. Lindenstrauss and V. Milman, editors, Geometric Aspects of Functional Analysis, pages 25–39. Birkhäuser Basel, Basel, 1995.

20

Jean Bourgain. On large values estimates for dirichlet polynomials and the density hypothesis for the riemann zeta function. International Mathematics Research Notices, 2000(3):133–146, 2000.

21

Jean Bourgain, Ciprian Demeter, and Larry Guth. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2), 184(2):633–682, 2016.

22

Joerg Bruedern and Trevor D. Wooley. On waring’s problem for larger powers, 2022.

23

D. A. Burgess. On character sums and L-series. Proc. London Math. Soc. (3), 12:193–206, 1962.

24

D. A. Burgess. On character sums and L-series. II. Proc. London Math. Soc. (3), 13:524–536, 1963.

25

F. Carlson. Über die nullstellen der dirichletschen reihen und der riemannschen ζ-funktion. Ark. Mat. Astron. Fys., 15(20):28, 1921.

26

Bin Chen, Gregory Debruyne, and Jasson Vindas. On the density hypothesis for l-functions associated with holomorphic cusp forms. Revista Matemática Iberoamericana, April 2024. Published online first.

27

Jing-run Chen. On the divisor problem for d3(n). Sci. Sinica, 14:19–29, 1965.

28

Jing Run Chen. On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet’s L-functions. Sci. Sinica, 20(5):529–562, 1977.

29

Jing Run Chen. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s L-functions. II. Sci. Sinica, 22(8):859–889, 1979.

30

Jing Run Chen and Jian Min Liu. On the least prime in an arithmetical progression. III. Sci. China Ser. A, 32(6):654–673, 1989.

31

Tsung-tao Chih. A divisor problem. Acad. Sinica Sci. Record, 3:177–182, 1950.

32

Laura Cladek and Terence Tao. Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle. Ars Inven. Anal., pages Paper No. 1, 38, 2021.

33

J. B. Conrey. At least two fifths of the zeros of the Riemann zeta function are on the critical line. Bulletin (New Series) of the American Mathematical Society, 20(1):79–81, 1989.

34

R. J. Cook. On the occurrence of large gaps between prime numbers. Glasgow Math. J., 20(1):43–48, 1979.

35

J. G. v. d. Corput. Verschärfung der Abschätzung beim Teilerproblem. Mathematische Annalen, 87(1-2):39–65, March 1922.

36

J. G. v. d. Corput. Zum Teilerproblem. Mathematische Annalen, 98(1):697–716, March 1928.

37

H. Cramér. über das Teilerproblem von Piltz. Arkiv för Mat. Astr. och. Fysik, 16(21), 1922.

38

H. Cramér. On the order of magnitude of the difference between consecutive prime numbers. Acta Arith., 2:23–46, 1936.

39

G. Csordas, T. S. Norfolk, and R. S. Varga. A lower bound for the de Bruijn-Newman constant Λ. Numer. Math., 52(5):483–497, 1988.

40

G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga. A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant Λ. Electron. Trans. Numer. Anal., 1:104–111 (electronic only), 1993.

41

G. Csordas, A. Ruttan, and R. S. Varga. The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis. Numerical Algorithms, 1(2):305–329, June 1991.

42

George Csordas, Wayne Smith, and Richard S. Varga. Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis. Constr. Approx., 10(1):107–129, 1994.

43

H. Davenport. On waring’s problem for fourth powers. Annals of Mathematics, 40(4):731–747, 1939.

44

H. Davenport. On waring’s problem for cubes. Acta Mathematica, 71(0):123–143, 1939.

45

N. G. de Bruijn. The roots of trigonometric integrals. Duke Math. J., 17:197–226, 1950.

46

Ciprian Demeter, Larry Guth, and Hong Wang. Small cap decouplings. Geometric and Functional Analysis, 30(4):989–1062, 08 2020.

47

Alexander Dobner. A proof of Newman’s conjecture for the extended Selberg class. Acta Arith., 201(1):29–62, 2021.

48

A K Dubitskas. A lower bound for the quantity ||(3/2)k||. Russian Mathematical Surveys, 45:163 – 164, 1990.

49

Johann Duttlinger and Wolfgang Schwarz. über die Verteilung der pythagoräischen Dreiecke. Colloq. Math., 43(2):365–372, 1980.

50

Jesse Elliott. Analytic number theory and algebraic asymptotic analysis. 2024.

51

P Erdös. On the arithmetical density of the sum of two sequences one of which forms a basis for the integers. Acta Arithmetica, 1(2):197–200, 1935.

52

E. Fogels. On the zeros of L-functions. Acta Arith., 11:67–96, 1965.

53

K. Ford. Vinogradov’s integral and bounds for the Riemann zeta function. Proc. London Math. Soc. (3), 85(3):565–633, 2002.

54

M. Forti and C. Viola. On large sieve type estimates for the Dirichlet series operator. In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), volume Vol. XXIV of Proc. Sympos. Pure Math., pages 31–49. Amer. Math. Soc., Providence, RI, 1973.

55

John Friedlander and Henryk Iwaniec. The Brun-Titchmarsh theorem. In Analytic number theory (Kyoto, 1996), volume 247 of London Math. Soc. Lecture Note Ser., pages 85–93. Cambridge Univ. Press, Cambridge, 1997.

56

P. X. Gallagher. A large sieve density estimate near σ=1. Invent. Math., 11:329–339, 1970.

57

Dorian M. Goldfeld. A further improvement of the Brun-Titchmarsh theorem. J. London Math. Soc. (2), 11(4):434–444, 1975.

58

S. Graham. An asymptotic estimate related to Selberg’s sieve. J. Number Theory, 10(1):83–94, 1978.

59

S. Graham. On Linnik’s constant. Acta Arith., 39(2):163–179, 1981.

60

Sidney West Graham. APPLICATIONS OF SIEVE METHODS. ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–University of Michigan.

61

Larry Guth and James Maynard. New large value estimates for dirichlet polynomials, 2024.

62

G. Halász. On the average of multiplicative number-theoretic functions. Acta Math. Hungar., 19:365–404, 1968.

63

G. Halász and P. Turán. On the distribution of roots of Riemann zeta and allied functions, I. Journal of Number Theory, 1(1):121–137, January 1969.

64

W. Haneke. Verschärfung der abschätzung von ζ(1/2+it). Acta Arithmetica, 8(4):357–430, 1963.

65

G. H. Hardy. On Dirichlet’s Divisor Problem. Proc. London Math. Soc. (2), 15:1–25, 1916.

66

G. H. Hardy. The Average Order of the Arithmetical Functions p(x) and δ(x). Proceedings of the London Mathematical Society, s2-15(1):192–213, 1917.

67

G. H. Hardy and J. E. Littlewood. Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes. Acta Mathematica, 41(0):119–196, 1916.

68

G. H. Hardy and J. E. Littlewood. The Approximate Functional Equation in the Theory of the Zeta-Function, with Applications to the Divisor-Problems of Dirichlet and Piltz. Proceedings of the London Mathematical Society, s2-21(1):39–74, 1923.

69

G. H. Hardy and J. E. Littlewood. On lindelöf’s hypothesis concerning the riemann zeta-function. Proc. R. Soc. A, pages 403–412, 1923.

70

G. H. Hardy and J. E. Littlewood. Some problems of ?partitio numerorum? (vi): Further researches in waring’s problem. Mathematische Zeitschrift, 23(1):1–37, December 1925.

71

G. Harman. Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2007.

72

Glyn Harman. Primes in short intervals. Math. Z., 180(3):335–348, 1982.

73

Glyn Harman. On the distribution of αp modulo one. J. London Math. Soc. (2), 27(1):9–18, 1983.

74

D. R. Heath-Brown. The differences between consecutive primes. J. London Math. Soc. (2), 18(1):7–13, 1978.

75

D. R. Heath-Brown. The twelfth power moment of the riemann-function. The Quarterly Journal of Mathematics, 29(4):443–462, 1978.

76

D. R. Heath-Brown. The differences between consecutive primes. II. J. London Math. Soc. (2), 19(2):207–220, 1979.

77

D. R. Heath-Brown. The differences between consecutive primes. III. J. London Math. Soc. (2), 20(2):177–178, 1979.

78

D. R. Heath-Brown. A Large Values Estimate for Dirichlet Polynomials. Journal of the London Mathematical Society, s2-20(1):8–18, August 1979.

79

D. R. Heath-Brown. Zero Density Estimates for the Riemann Zeta-Function and Dirichlet L -Functions. Journal of the London Mathematical Society, s2-19(2):221–232, April 1979.

80

D. R. Heath-Brown. Mean values of the zeta-function and divisor problems. In Recent Progress in Analytic Number Theory, volume 1, pages 115–119, Durham 1979, 1981. Academic, London.

81

D. R. Heath-Brown. Finding primes by sieve methods. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 487–492. PWN, Warsaw, 1984.

82

D. R. Heath-Brown. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. (3), 64(2):265–338, 1992.

83

D. R. Heath-Brown. A new kth derivative estimate for exponential sums via Vinogradov’s mean value. Proceedings of the Steklov Institute of Mathematics, 296(1):88–103, January 2017.

84

D. R. Heath-Brown and H. Iwaniec. On the difference between consecutive primes. Invent. Math., 55(1):49–69, 1979.

85

Roger Heath-Brown. The Differences Between Consecutive Primes, V. International Mathematics Research Notices, 2021(22):17514–17562, November 2021.

86

Hans Heilbronn. über den Primzahlsatz von Herrn Hoheisel. Math. Z., 36(1):394–423, 1933.

87

Harald Andres Helfgott. The ternary goldbach problem, 2015.

88

G. Hoheisel. Primzahlprobleme in der analysis. Siz. Preuss. Akad. Wiss., 33:580–588, 1930.

89

M. N. Huxley. On the difference between consecutive primes. Invent. Math., 15:164–170, 1972.

90

M. N. Huxley. A note on large gaps between prime numbers. Acta Arith., 38(1):63–68, 1980/81.

91

M. N. Huxley. Exponential Sums and the Riemann Zeta Function IV. Proceedings of the London Mathematical Society, s3-66(1):1–40, January 1993.

92

M. N. Huxley. Area, Lattice Points, and Exponential Sums. Number New Ser., 13 in Oxford Science Publications. Clarendon Press; Oxford University Press, Oxford; New York, 1996.

93

M. N. Huxley. Exponential Sums and Lattice Points III. Proceedings of the London Mathematical Society, 87(03):591–609, November 2003.

94

M. N. Huxley. Exponential sums and the Riemann zeta function v. Proceedings of the London Mathematical Society, 90(01):1–41, January 2005.

95

M. N. Huxley and G. Kolesnik. Exponential Sums and the Riemann Zeta Function III. Proceedings of the London Mathematical Society, s3-62(3):449–468, May 1991.

96

M. N. Huxley and G. Kolesnik. Exponential Sums and the Riemann Zeta Function III. Proceedings of the London Mathematical Society, s3-66(2):302–302, March 1993.

97

M. N. Huxley and N. Watt. Exponential Sums and the Riemann Zeta Function. Proceedings of the London Mathematical Society, s3-57(1):1–24, July 1988.

98

M. N. Huxley and N. Watt. The Hardy-Littlewood method for exponential sums. In Number theory, Vol. I (Budapest, 1987), volume 51 of Colloq. Math. Soc. János Bolyai, pages 173–191. North-Holland, Amsterdam, 1990.

99

Martin Huxley. Large values of Dirichlet polynomials. Acta Arithmetica, 24(4):329–346, 1973.

100

Martin Huxley. Large values of Dirichlet polynomials II. Acta Arithmetica, 27:159–170, 1975.

101

Martin Huxley. Large values of Dirichlet polynomials, III. Acta Arithmetica, 26(4):435–444, 1975.

102

Martin N. Huxley and Grigori Kolesnik. Exponential sums with a large second derivative. In Matti Jutila and Tauno Metsänkylä, editors, Number Theory, pages 131–144. De Gruyter, Berlin, Boston, January 2001.

103

A. E. Ingham. On the difference between consecutive primes. The Quarterly Journal of Mathematics, os-8(1):255–266, 1937.

104

A. E. Ingham. On the estimation of N(σ,t). The Quarterly Journal of Mathematics, os-11(1):201–202, 1940.

105

A. E. Ingham. The distribution of prime numbers. Number no. 30 in Cambridge mathematical library. Cambridge University Press, Cambridge ; New York, 1990.

106

A. Ivić. Topics in Recent Zeta Function Theory. Publications mathématiques d’Orsay. Université de Paris-Sud, Département de Mathématique, 1983.

107

A. Ivić. A zero-density theorem for the Riemann zeta-function. Tr. Mat. Inst. Steklova, 163:85–89, 1984.

108

Aleksandar Ivić. On sums of large differences between consecutive primes. Math. Ann., 241(1):1–9, 1979.

109

Aleksandar Ivić. Exponent pairs and the zeta function of Riemann. Studia Sci. Math. Hungar., 15(1-3):157–181, 1980.

110

Aleksandar Ivić. The Riemann zeta-function. Dover Publications, Inc., Mineola, NY, 2003. Theory and applications, Reprint of the 1985 original [Wiley, New York; MR0792089 (87d:11062)].

111

Aleksandar Ivic and Michel Ouellet. Some new estimates in the dirichlet divisor problem. Acta Arithmetica, 52(3):241–253, 1989.

112

Aleksandar Ivić. A note on the zero-density estimates for the zeta function. Archiv der Mathematik, 33(1):155–164, December 1979.

113

H. Iwaniec and E. Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.

114

H Iwaniec and C.J Mozzochi. On the divisor and circle problems. Journal of Number Theory, 29(1):60–93, May 1988.

115

Henryk Iwaniec. On the Brun-Titchmarsh theorem. J. Math. Soc. Japan, 34(1):95–123, 1982.

116

Henryk Iwaniec and Matti Jutila. Primes in short intervals. Ark. Mat., 17(1):167–176, 1979.

117

Henryk Iwaniec and János Pintz. Primes in short intervals. Monatsh. Math., 98(2):115–143, 1984.

118

Olli Järviniemi. On large differences between consecutive primes, 2022.

119

Chao Hua Jia. Goldbach numbers in short interval. I. Sci. China Ser. A, 38(4):385–406, 1995.

120

Chao Hua Jia. On the exceptional set of Goldbach numbers in a short interval. Acta Arith., 77(3):207–287, 1996.

121

Chaohua Jia. Difference between consecutive primes. Sci. China Ser. A, 38(10):1163–1186, 1995.

122

Chaohua Jia. Almost all short intervals containing prime numbers. Acta Arith., 76(1):21–84, 1996.

123

Chen Jing-run. On the least prime in an arithmetical progression. Sci. Sinica, 14:1868–1871, 1965.

124

M. Jutila. Zeros of the zeta-function near the critical line. In Studies in pure mathematics, pages 385–394. Birkhäuser, Basel, 1983.

125

Matti Jutila. A new estimate for Linnik’s constant. Ann. Acad. Sci. Fenn. Ser. A I, 471:8, 1970.

126

Matti Jutila. On Linnik’s constant. Math. Scand., 41(1):45–62, 1977.

127

Matti Jutila. On Linnik’s constant. Math. Scand., 41(1):45–62, 1977.

128

Matti Jutila. Zero-density estimates for L-functions. Acta Arithmetica, 32(1):55–62, 1977.

129

Koichi Kawada and Trevor D. Wooley. On the waring–goldbach problem for fourth and fifth powers. Proceedings of the London Mathematical Society, 83(1):1–50, 2001.

130

Bryce Kerr. Large values of dirichlet polynomials and zero density estimates for the riemann zeta function, 2019.

131

Haseo Ki, Young-One Kim, and Jungseob Lee. On the de Bruijn-Newman constant. Adv. Math., 222(1):281–306, 2009.

132

Henry Kim and Peter Sarnak. Refined estimates towards the ramanujan and selberg conjectures. J. Amer. Math. Soc, 16(1):175–181, 2003.

133

G. Kolesnik. On the estimation of multiple exponential sums. In Recent progress in analytic number theory, Vol. 1 (Durham, 1979), pages 231–246. Academic Press, London-New York, 1981.

134

G. Kolesnik. On the order of ζ(12+it) and Δ(R). Pacific Journal of Mathematics, 98(1):107–122, 1982.

135

G. Kolesnik. On the method of exponent pairs. Acta Arith., 45(2):115–143, 1985.

136

G. A. Kolesnik. The improvement of the remainder term in the divisor problem. Mat. Zametki, 6:545–554, 1969.

137

G. A. Kolesnik. An estimate for certain trigonometric sums. Acta Arith., 25:7–30. (errata insert), 1973/74.

138

Angel V. Kumchev and Trevor D. Wooley. On the waring–goldbach problem for seventh and higher powers. Monatshefte für Mathematik, 183(2):303–310, June 2016.

139

J. Lambek and L. Moser. On the distribution of Pythagorean triangles. Pacific J. Math., 5:73–83, 1955.

140

Edmund Landau. über die anzahl der gitterpunkte in geweissen bereichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1912:687–770, 1912.

141

Andrew V. Lelechenko. Linear programming over exponent pairs. Acta Universitatis Sapientiae, Informatica, 5(2):271–287, 2014.

142

Hong Ze Li. Goldbach numbers in short intervals. Sci. China Ser. A, 38(6):641–652, 1995.

143

Hongze Li. Primes in short intervals. Math. Proc. Cambridge Philos. Soc., 122(2):193–205, 1997.

144

Runbo Li. Primes in almost all short intervals, 2024.

145

Runbo Li. The number of primes in short intervals and numerical calculations for harman’s sieve, 2025.

146

Xiaochun Li and Xuerui Yang. An improvement on gauss’s circle problem and dirichlet’s divisor problem, 2023.

147

Linnik. An elementary solution of the problem of waring by schnirelman’s method. Matematiceskij sbornik, 54(2):225–230, 1943.

148

U. V. Linnik. On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. [Mat. Sbornik] N.S., 15/57:139–178, 1944.

149

U. V. Linnik. On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S., 15/57:347–368, 1944.

150

Yu. V. Linnik. On Erdős’s theorem on the addition of numerical sequences. Mat. Sb., Nov. Ser., 10:67–78, 1942.

151

Yu. V. Linnik. On the representation of large numbers as sums of seven cubes. Mat. Sb., Nov. Ser., 12:218–224, 1943.

152

Jianya Liu, Trevor D Wooley, and Gang Yu. The quadratic waring–goldbach problem. Journal of Number Theory, 107(2):298–321, 2004.

153

Shi Tuo Lou and Qi Yao. On the upper bound of difference between consecutive primes. Kexue Tongbao, 30(8):1127–1128, 1985.

154

Shi Tuo Lou and Qi Yao. On the Brun-Titchmarsh theorem. Ziran Zazhi, 5(393), 1986.

155

Shi Tuo Lou and Qi Yao. Upper bounds for primes in intervals. Chinese Ann. Math. Ser. A, 10(3):255–262, 1989.

156

Shi Tuo Lou and Qi Yao. A Chebychev’s type of prime number theorem in a short interval. II. Hardy-Ramanujan J., 15:1–33, 1992.

157

Shituo Lou and Qi Yao. The number of primes in a short interval. Hardy-Ramanujan Journal, 16:127, January 1993.

158

Wenhui Lu and Hengjie Yuan. Primes in a short interval, 2025.

159

K. Mahler. On the fractional parts of the powers of a rational number (ii). Mathematika, 4(2):122–124, 1957.

160

Helmut Maier. Primes in short intervals. Michigan Math. J., 32(2):221–225, 1985.

161

K. Matomäki. Large differences between consecutive primes. The Quarterly Journal of Mathematics, 58(4):489–518, August 2007.

162

Kaisa Matomäki and Joni Teräväinen. A note on zero density results implying large value estimates for dirichlet polynomials, 2024.

163

J. Maynard. On the difference between consecutive primes, 2012.

164

James Maynard. On the Brun-Titchmarsh theorem. Acta Arith., 157(3):249–296, 2013.

165

Zaizhao Meng. The distribution of the zeros of L-functions and the least prime in some arithmetic progression. Sci. China Ser. A, 43(9):937–944, 2000.

166

Hartmut Menzer. On the number of primitive Pythagorean triangles. Math. Nachr., 128:129–133, 1986.

167

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168

H. L. Montgomery. Zeros of L-functions. Invent. Math., 8:346–354, 1969.

169

H. L. Montgomery and R. C. Vaughan. The large sieve. Mathematika, 20:119–134, 1973.

170

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171

Yoichi Motohashi. On some improvements of the Brun-Titchmarsh theorem. II. Sūrikaisekikenkyūsho Kokyūroku, (193):97–109, 1973.

172

Yoichi Motohashi. On some improvements of the Brun-Titchmarsh theorem. J. Math. Soc. Japan, 26:306–323, 1974.

173

C. J. Mozzochi. On the difference between consecutive primes. J. Number Theory, 24(2):181–187, 1986.

174

Charles M. Newman. Fourier transforms with only real zeros. Proc. Amer. Math. Soc., 61(2):245–251 (1977), 1976.

175

Charles M. Newman and Wei Wu. Constants of de Bruijn–Newman type in analytic number theory and statistical physics. Bull. Amer. Math. Soc. (N.S.), 57(4):595–614, 2020.

176

T. S. Norfolk, A. Ruttan, and R. S. Varga. A lower bound for the de Bruijn-Newman constant Λ. II. In Progress in approximation theory (Tampa, FL, 1990), volume 19 of Springer Ser. Comput. Math., pages 403–418. Springer, New York, 1992.

177

A. M. Odlyzko. An improved bound for the de Bruijn-Newman constant. volume 25, pages 293–303. 2000. Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999).

178

Cheng Dong Pan. On the least prime in an arithmetical progression. Sci. Record (N.S.), 1:311–313, 1957.

179

Cheng Dong Pan. On the least prime in an arithmetical progression. Acta. Sci. Natur. Univ. Pekinensis, 4:1–34, 1958.

180

A. S. Peck. On the differences between consecutive primes. PhD thesis, University of Oxford, 1996.

181

A. S. Peck. Differences Between Consecutive Primes. Proceedings of the London Mathematical Society, 76(1):33–69, January 1998.

182

Eric Phillips. The zeta-function of riemann; further developments of van der corput’s method. The Quarterly Journal of Mathematics, os-4(1):209–225, 1933.

183

J. Pintz. On primes in short intervals. I. Studia Sci. Math. Hungar., 16(3-4):395–414, 1981.

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